So I got Touhou 7.5: Immaterial and Missing Power, but suddenly I've been thrust into a world where 12-year old girls wearing strange outfits and wielding dangerous weapons scream japanese at each other before firing off gratuitous amounts of magical projectiles in all directions. What's going on here? What is the plotline of these games?

While making preparations for the feast three days away, Reimu is distracted by a strange powerful mist that is covering Gensokyo. She and many other inhabitants of Gensokyo (such as Alice and Patchouli) go to investigate.

Suika Ibuki is an oni. Although legends say all the oni are gone, Suika has decided to reunite the humans and oni by attracting them through partying. Using her special ability to spread across the land, she causes the humans and youkai to have feasts and parties. The more drunk she gets, the stronger she becomes, and she carries a gourd treasured by the oni, which never empties of sake.

Still pretty bizarre, but then, that's practically the definition of anime...

Note that there are further dialogue translations provided in that first link, that will probably prove helpful if you really want to try and understand the details as the story unfolds. Good luck with that. ;)

Spring has passed by the land of Gensokyo, a reclusive realm in the far east, and the cherry trees have since shed their blossoms. However, the hanami kept on going, with feasts being hosted day after day with no end in sight. Adding onto that, every time the feast is held, an unknown restless spiritual aura in Gensokyo also increases; however, nothing happens even while the spiritual aura rises, but when the spiritual aura rises, not a single person attempts to stop the feasts. As such, everyone who goes to the feast, be it human or yōkai, appear to be very suspicious.

Three days before the next feast, the heroines set out, each on their own, and attempt to investigate.

"I'll get to the bottom of this before the next feast!"

The Touhou Project Series is known for their signature Bullet Hell gameplay, the developpers have incorporated this type of gameplay in a fighting game instead of a shoot'm up, by focussing on projectiles